There are many “paradoxical sets” of reals that can be obtained using a well-ordering of the reals, which is a consequence of the Axiom of Choice. In \({\rm ZF}\), can we recover the well-ordering of the reals from the existence of a given paradoxical set? Under certain conditions of Extendability and Amalgamation, we give a negative answer to this question. In particular, we solve it for the paradoxical set given by a partition of \(\mathbb{R}^3\) in unit circles. For this, some geometrical and algebraic considerations are needed.
